There is comp, when everything looks/works fine. But when try to nest this "horse" comp into another or try to render it, file and preview remains empty (no image, black, nothing).
But, when I try to "Save frame as" - image is there.

Your layers are all guide layers -- meaning they are visible in the viewer for the comp they are in, but they will not render on output or when precomposed. This is a great feature when you use it intentionally to bring some kind of reference into a comp, but it can be quite confusing if you've set them accidentally!

To fix your issue, select your horse layers, then from the Layer menu, toggle "Guide Layer." They will now render normally.